The literature on adjustment costs in changing the level of capital is very large. The following papers and their references point you to that literature.

One way of thinking about this literature is to focus on the capital accumulation equation, K(t+1) = (1-delta)*K(t) + I(t). Here, there are no adjustment costs. Adjustment costs refer to a situation in which there is curvature relating I(t) to K(t+1). There are two ways in which this curvature has been modeled. The traditional way is with the addition of –phi(I(t)/K(t))*K(t) to the above expression, where phi is a concave function with its minimum at the steady state of the investment-capital ratio. Another is with the addition of –phi(I(t)/I(t-1))*I(t) to the above term. Here, the adjustment costs are in terms of the change in the flow of investment. The latter was proposed in Christiano, Eichenbaum and Evans (JPE, 2004) (CEE). Another model, the time to build model, was proposed in Kydland and Prescott’s 1982 Econometrica paper. Here are some papers related to this literature:

Christiano and Todd, arguing that micro data on investment projects suggests a modification, ‘time-to-plan’, on Kydland and Prescott’s time to build model.

Christiano and Vigfusson, describing a spectral approximation to the Gaussian likelihood for estimating the time to plan model.

David Lucca (see first chapter), describing a micro-founded way of thinking of the CEE model of adjustment costs.

Kiminori Matsuyama, writing in 1984, describing another micro-founded way to obtain the CEE formulation of adjustment costs.

Miles Kimball on adjustment costs.

Boldrin, Christiano and Fisher, on adjustment costs in an equilibrium model that tries to not just come to terms with business cycles, but also asset prices.